Preheat the oven to 350°F (177°C). Line two baking sheets with parchment paper.

Measure 1/4 cup rainbow sprinkles. Reserve about 1 tablespoon for topping and set it aside; add the remaining sprinkles to the dough in a later step.

If you use salted butter, omit the 1/2 teaspoon salt from the recipe.

In a large bowl, place 1 cup unsalted butter (room temperature) and 1 cup granulated sugar. Using an electric mixer, cream together on medium speed until light and fluffy, about 2–3 minutes.

Add 2 large eggs and mix on medium speed until fully incorporated.

Add 1/2 teaspoon salt (unless you used salted butter), 1 teaspoon vanilla extract, and 1 1/2 teaspoons cake batter flavoring. Mix briefly to combine.

Add 2 teaspoons baking powder and 3 cups all-purpose flour. Mix on low speed just until the flour is incorporated and a dough forms. Do not overmix.

Fold the measured portion of rainbow sprinkles (the portion not reserved for topping) into the dough by hand until evenly distributed.

Portion the dough into 12 equal balls using a 1/4-cup measuring cup (or about 75 grams per ball). Place six dough balls on each prepared baking sheet, spacing them about 2 inches apart.

Press each dough ball gently with a flat-bottomed glass to spread them to just under 1/4 inch thickness.

Bake at 350°F for 11–12 minutes for a slightly underbaked center. Add 1–2 minutes if you prefer firmer, more well-done cookies. Remove from oven and let cookies cool on the baking sheet for 2 minutes.

Transfer the cookies to a wire cooling rack and cool completely before frosting.

Make the frosting: in a medium mixing bowl, beat 4 ounces softened cream cheese and 3 tablespoons softened butter until smooth. Add 2 cups powdered sugar, 3 tablespoons milk, and 1/2 teaspoon cake batter flavoring. Beat until the frosting is smooth and spreadable.

Fit a piping bag with a medium tip, fill it with frosting, and pipe a pinwheel on each cooled cookie, starting from the center and working outward. Immediately top each frosted cookie with the reserved sprinkles.

Allow the frosted cookies to sit for about 30 minutes to let the frosting firm before serving.